Optimal. Leaf size=181 \[ -\frac{\left (\sqrt{13} \left (568 \sqrt{13} m-1168 m+1701\right )+1521\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{338 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{\left (\sqrt{13} (1701-1168 m)-13 (568 m+117)\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{338 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(209-426 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.280688, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1648, 830, 68} \[ -\frac{\left (\sqrt{13} \left (568 \sqrt{13} m-1168 m+1701\right )+1521\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{338 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{\left (\sqrt{13} (1701-1168 m)-13 (568 m+117)\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{338 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(209-426 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1648
Rule 830
Rule 68
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3 (1+4 x)^m}{\left (1-5 x+3 x^2\right )^2} \, dx &=\frac{(209-426 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{1}{507} \int \frac{(1+4 x)^m (13 (1143+836 m)-39 (117+568 m) x)}{1-5 x+3 x^2} \, dx\\ &=\frac{(209-426 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{1}{507} \int \left (\frac{\left (-39 (117+568 m)-3 \sqrt{13} (-1701+1168 m)\right ) (1+4 x)^m}{-5-\sqrt{13}+6 x}+\frac{\left (-39 (117+568 m)+3 \sqrt{13} (-1701+1168 m)\right ) (1+4 x)^m}{-5+\sqrt{13}+6 x}\right ) \, dx\\ &=\frac{(209-426 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{1}{169} \left (\sqrt{13} (1701-1168 m)-13 (117+568 m)\right ) \int \frac{(1+4 x)^m}{-5-\sqrt{13}+6 x} \, dx+\frac{1}{169} \left (\sqrt{13} (1701-1168 m)+13 (117+568 m)\right ) \int \frac{(1+4 x)^m}{-5+\sqrt{13}+6 x} \, dx\\ &=\frac{(209-426 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{\left (\sqrt{13} (1701-1168 m)+13 (117+568 m)\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13-2 \sqrt{13}}\right )}{338 \left (13-2 \sqrt{13}\right ) (1+m)}+\frac{\left (\sqrt{13} (1701-1168 m)-13 (117+568 m)\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13+2 \sqrt{13}}\right )}{338 \left (13+2 \sqrt{13}\right ) (1+m)}\\ \end{align*}
Mathematica [A] time = 0.299516, size = 252, normalized size = 1.39 \[ \frac{(4 x+1)^{m+1} \left (-\frac{12 \left (\sqrt{13} (1215-292 m)+1846 m\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )}{\left (13-2 \sqrt{13}\right ) (m+1)}-\frac{351 \left (27 \sqrt{13}-13\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )}{\left (2 \sqrt{13}-13\right ) (m+1)}+\frac{12 \left (\sqrt{13} (1215-292 m)-1846 m\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )}{\left (13+2 \sqrt{13}\right ) (m+1)}-\frac{351 \left (13+27 \sqrt{13}\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )}{\left (13+2 \sqrt{13}\right ) (m+1)}+\frac{5434-11076 x}{3 x^2-5 x+1}\right )}{1014} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.35, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 4\,x+1 \right ) ^{m} \left ( 2+3\,x \right ) ^{3}}{ \left ( 3\,{x}^{2}-5\,x+1 \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (4 \, x + 1\right )}^{m}{\left (3 \, x + 2\right )}^{3}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}{\left (4 \, x + 1\right )}^{m}}{9 \, x^{4} - 30 \, x^{3} + 31 \, x^{2} - 10 \, x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (4 \, x + 1\right )}^{m}{\left (3 \, x + 2\right )}^{3}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]